Daily crude oil production by Pemex, Mexico's national oil company, for 2008–2013 could be approximated by

Question

P(t) = 0.017t² − 0.4t + 5.23 million barrels  (8 ≤ t ≤ 13)

where t is time in years since the start of 2000.† Find the derivative function dP/dt

dP/dt=

At what rate was oil production changing at the start of 2009 (t = 9)?

Daily oil production was decreasing at a rate of ____  million barrels per year.

Answer 1

Hi Alice. If we understand the meaning of derivatives and how to find them, this problem is relatively straightforward.

Recall that a derivative will always refer to the rate with which something changes at a given instant in time. The function we are given represents crude oil production in millions of barrels, however it asks us to find millions of barrels per year in 2009. This is a rate of change and it specifies the instant in time which we are to find it, in this case a particular year.

To find the derivative of a function, we must look at each term individually, decrease its exponent by 1, and multiply by the old exponent out front. Performing this on each term, we get:

dP/dt = 2(0.017t) - 0.4, or

dP/dt = 0.034t - 0.4

This gives us the function to find the rate of change from 2008-2013, since we are told that the function is bound between t = 8 and t = 13. To find the rate of change for 2009, we substitute t = 9:

dP/dt = 0.034(9) - 0.4

dP/dt = -0.094

Since we ended up with a negative answer for rate of change, we know it is decreasing. So in 2009, daily oil production was decreasing at a rate of 0.094 million barrels per year.

Hope this helps!

Answer 2